WARMUP WRITE DOWN 5 WAYS IN WHICH YOU SEE/USE PROBABILITY IN EVERY DAY LIFE.
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Transcript of WARMUP WRITE DOWN 5 WAYS IN WHICH YOU SEE/USE PROBABILITY IN EVERY DAY LIFE.
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WARMUP
•WRITE DOWN 5 WAYS IN WHICH YOU SEE/USE PROBABILITY IN EVERY DAY LIFE.
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CHAPTER 3: PROBABILITY
3.1 BASIC CONCEPTS OF PROBABILITY AND COUNTING
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VOCABULARY • PROBABILITY EXPERIMENT: AN ACTION, OR TRIAL, THROUGH
WHICH SPECIFIC RESULTS (COUNTS, MEASURES, OR RESPONSES) ARE OBTAINED.
• OUTCOME: THE RESULT OF A SINGLE TRIAL OF A PROBABILITY EXPERIMENT
• SAMPLE SPACE: THE SET OF ALL POSSIBLE OUTCOMES IN A SAMPLE SPACE. EVENTS ARE SUBSETS OF THE SAMPLE SPACE.
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DETERMINE THE NUMBER OF OUTCMES AND IDENTIFY THE SAMPLE SPACE
• DRAWING A DIAMOND OUT OF A DECK OF CARDS
• ROLLING A 2 ON A 6-SIDED DIE.
• DRAWING A KING OUT OF A DECK OF CARDS
• FLIPPING HEADS ON A COIN
• DRAWING A BLACK 4 FROM A DECK OF CARDS.
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SOMETIMES WE CAN DRAW OUT THE NUMBER OF RESPONSES USING A DIAGRAM
• SUPPOSE WE PROPOSED THE QUESTION:
• DOES YOUR FAVORITE TEAM’S WIN OR LOSS AFFECT YOUR MOOD?
• A. YES B. NO. C. NOT SURE
• LET’S SAY WE WANTED TO DIVIDE THE RESPONSES UP BY GENDER AS WELL. WHAT WOULD OUR SAMPLE SPACE LOOK LIKE? HOW BIG WOULD IT BE?
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THE FUNDAMENTAL COUNTING PRINCIPLE
• IF EVENT 1 CAN OCCUR M WAYS AND EVENT 2 CAN OCCUR N WAYS, THEN BOTH EVENTS CAN OCCUR A NUMBER OF M X N WAYS. WE USE THIS RULE FOR EVENTS OCCURRING IN SEQUENCE.
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EXAMPLE…
• SUPPOSE YOU TOOK YOUR DATE OUT FOR A NICE STEAK DINNER. YOU HAVE THE FOLLOWING OPTIONS FOR YOUR CHOICE OF STEAK AND A SIDE.
• CUT OF STEAK: RIBEYE, T-BONE, NEW YORK STRIP, FILET MIGNON, PORTERHOUSE
• TEMPERATURE: RARE, MEDIUM RARE, MEDIUM WELL, WELL-DONE
• SIDE: LOADED BAKE POTATOES, SWEET POTATOES, RICE PILAF, MACORONI AND CHEESE, SALAD.
• USE THE FUNDAMENTAL COUNTING PRINCIPLE TO FIND OUT HOME MANY COMBINATIONS YOU CAN MAKE.
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YOUR BANK ACCOUNT HAS A 4-DIGIT PIN NUMBER…
• HOW MANY POSSIBLE CODES ARE THERE IF YOU CAN REPEAT EVERY NUMBER?
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PIN NUMBER CONTINUED…
• WHAT IF NUMBERS COULD NOT BE REPEATED AND ONLY USED ONCE?
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CLASSICAL PROBABILITY
• CLASSICAL PROBABILITY IS USED WHEN EACH OUTCOME IN A SAMPLE IS EQUALLY LIKELY TO OCCUR.
• P(E)=
• WHAT IS THE PROBABILITY OF ROLLING A THREE ON A SIX SIDED DIE?
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SUPPOSE YOU ROLLED A SIX-SIDED DIE
• WHAT IS THE PROBABILITY YOU ROLL AN EVEN NUMBER?
• A NUMBER GREATER THAN 4?
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WHAT IS THE PROBABILITY OF LANDING ON A RED SPACE? (THERE ARE 24 SPACES)
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EMPIRICAL PROBABILITY
• EMPIRICAL PROBABILITY: BASED ON OBSERVATIONS OBTAINED FROM PROBABILITY EXPERIMENTS.
• THE FORMULA… P(E)=
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EXAMPLE…• SUPPOSE THE FOLLOWING FREQUENCY DISTRIBUTION REPRESENT GRADES
ON A TEST…
• FIND THE PROBABILITY THAT YOU MADE A C ON THE TEST…
• FIND THE PROBABILITY THAT YOU MADE AN A OR B ON THE TEST…GRADE FREQUENCY
A 5
B 7
C 16
D 4
F 3
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LAST PART OF 3.1• LAW OF LARGE NUMBERS: REPETITION OF AN EXPERIMENT SO THAT THE
EMPIRICAL PROBABILITY OF AN EVENT APPROACHES THE ACTUAL PROBABILITY OF THE EVENT.
• COMPLEMENT OF EVENT E: SET OF ALL OUTCOMES IN A SAMPLE SPACE THAT ARE NOT INCLUDED IN EVENT E. DENOTED AS E’.
• LOOKING BACK AT THE TEST SCORES TABLE. WHAT IS THE PROBABILITY THAT YOU DID NOT MAKE A C ON THE TEST?
GRADE FREQUENCY
A 5
B 7
C 16
D 4
F 3
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SECTION 3.2CONDITIONAL PROBABILITY AND THE MULTIPLICATION RULE
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CONDITIONAL PROBABILITY
• PROBABILITY OF AN EVENT OCCURING, GIVEN THAT ANOTHER HAS ALREADY OCCURRED.
• READ AS “PROBABILITY OF B, GIVEN A.” AND DENOTED BY P(B|A)
• YOU ARE DEALT TWO CARDS FROM A 52-CARD DECK, ONE FACE UP, AND ONE FACE DOWN. IF THE CARD SHOWING IS AN ACE, WHAT IS THE PROBABILITY THAT THE CARD FACE DOWN IS AN ACE AS WELL?
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EXAMPLE 2…• THE RESULTS OF A MIDDLE SCHOOL SURVEY ON FAVORITE WINTER SPORTS
ARE AS FOLLOWS:
• WHAT IS THE PROBABILITY OF SELECTING A 6TH GRADER?
• WHAT IS THE PROBABILITY OF SELECTING A 7TH GRADER THAT PREFERS SNOWBOARDING?
• SUPPOSE THE STUDENT SELECTED LIKES SKIING, WHAT IS THE PROBABILITY THAT IT IS AN 8TH GRADER?
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TYPES OF EVENTS AND THE MULTIPLICATION RULE
• TWO EVENTS ARE INDEPENDENT IF THE OCCURRENCE OF ONE EVENT DOES NOT AFFECT THE PROBABILITY OF THE SECOND EVENT. OTHERWISE THEY ARE SAID TO BE DEPENDENT EVENTS.
• INDEPENDENT EVENTS ARE DENOTED AS P(B|A)=P(B) OR IF P(A|B)=P(A)
• DETERMINE IF THE FOLLOWING ARE DEPENDENT OR INDEPENDENT EVENTS…
• SELECTING A KING FROM A DECK OF CARDS, REPLACING IT, AND THEN SELECTING AN ODD NUMBER.
• TOSSING A COIN AND THEN ROLLING A 6-SIDED DIE
• NOT DOING YOUR HOMEWORK AND TRYING TO PASS THE CLASS.
• WINNING THE VOLLEYBALL GAME AND THE WEATHER BEING SUNNY.
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MULTIPLICATION RULE
•PROBABILITY THAT TWO EVENTS WILL OCCUR IN SEQUENCE IS:
•P(A AND B)= P(A) X P(B|A). IF THE EVENTS ARE INDEPENDENT THE FORMULA CAN SIMPLY BE WRITTEN AS P(A) X P(B).
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EXAMPLES…• PROBABILITY OF GETTING A 10 DEALT TO YOU AND THEN
GETTING AN ACE DEALT TO YOU.
• P(10)=…. P(A)=…
• THE PROBABILITY THAT A WEIGHT LOSS PILL IS SUCCESSFUL IS 71%.
• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON 4 PEOPLE?
• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON 0 OF THE 4 PEOPLE?
• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON AT LEAST 1 OF THE 4 PEOPLE?
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THE ADDITION RULESECTION 3-3
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MUTUALLY EXCLUSIVE EVENTS
• EVENTS THAT OCCUR IN SEQUENCE HAVE A PROBABILITY DENOTED BY P(A AND B).
• MUTUALLY EXCLUSIVE EVENTS ARE DENOTED BY P(A OR B).
• TWO EVENTS ARE MUTUALLY EXCLUSIVE IF EVENT A AND EVENT B CANNOT OCCUR AT THE SAME TIME
• WE USE MUTUALLY EXCLUSIVE EVENTS TO FIND THE PROBABILITY THAT AT LEAST ONE EVENT WILL OCCUR.
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EXAMPLES…• DECIDE IF THE FOLLOWING EVENTS ARE MUTUALLY
EXCLUSIVE. CAN THEY OCCUR AT THE SAME TIME? DO THEY HAVE ANY OUTCOMES IN COMMON?
• EVENT A: SELECTING A JACK FROM A STANDARD DECK OF CARDS
• EVENT B: SELECTING A FACE CARD FROM A STANDARD DECK OF CARDS
• EVENT A: RANDOMLY SELECTING A VEHICLE THAT IS A FORD
• EVENT B: RANDOMLY SELECTING A VEHICLE THAT IS A TOYOTA
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THE ADDITION RULE
• THE PROBABILITY THAT EVENTS A OR BB WILL OCCUR P(A OR B) IS GEVEN BY
• P(A OR B)= P(A)+P(B)-P(A AND B)
• IF THEY ARE MUTUALLLY EXCLUSIVE, THEN OUR FORMULA IS SIMPLY P(A OR B)= P(A)+P(B)
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P(A OR B)= P(A)+P(B)-P(A AND B)
• YOU WIN $10 IF YOU ARE DEALT A QUEEN OR A HEART FROM A STANDARD DECK OF CARDS. FIND THE PROBABILITY THAT YOU WIN THE GAME.
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Blood Types
O A B AB TOTAL
Positive 156 139 37 12 344
Negative 28 25 8 4 65
Total 184 164 45 16 409
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TYPE OF PROBABILITY FORMULA/SYMBOLS
CLASSICAL PROBABILITY
EMPIRICAL PROBABILITY
RANGE OF PROBABILITIES RULE
COMPLMENTARY EVENTS
MULTIPLICATION RULE
ADDITION RULE